A relation among tangle, 3-tangle, and von Neumann entropy of entanglement for three qubits

TL;DR

This paper derives a comprehensive formula linking tangle, 3-tangle, and von Neumann entropy for three-qubit pure states, revealing their interrelations and identifying states with maximal entanglement properties.

Contribution

It introduces a general formula connecting tangle, 3-tangle, and von Neumann entropy, extending previous work by relating these measures through LU invariants and ASD.

Findings

Derived a general formula for tangle in three-qubit pure states.

Identified states with non-zero tangle, 3-tangle, and von Neumann entropy that remain entangled when one qubit is traced out.

Showed that the W state maximizes average von Neumann entropy within its SLOCC class.

Abstract

In this paper, we derive a general formula of the tangle for pure states of three qubits, and present three explicit local unitary (LU) polynomial invariants. Our result goes beyond the classical work of tangle, 3-tangle and von Neumann entropy of entanglement for Ac\'{\i}n et al.' Schmidt decomposition (ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann entropy for ASD with Ac\'{\i}n et al.'s LU invariants. In particular, our result reveals a general relation among tangle, 3-tangle, and von Neumann entropy, together with a relation among their averages. The relations can help us find the entangled states satisfying distinct requirements for tangle, 3-tangle, and von Neumann entropy. Moreover, we obtain all the states of three qubits of which tangles, concurrence, 3-tangle and von Neumann entropy don't vanish and these states are endurable when one of three qubits is traced out. We indicate that for the three-qubit W state, its average von Neumann entropy is maximal only within the W SLOCC class, and that under ASD the three-qubit GHZ state is the unique state of which the reduced density operator obtained by tracing any two qubits has the maximal von Neumann entropy.